Properties

Label 4-3920e2-1.1-c1e2-0-12
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 9-s + 6·11-s + 6·13-s − 4·15-s − 2·17-s − 12·19-s − 12·23-s + 3·25-s + 6·27-s − 6·29-s − 12·31-s − 12·33-s − 4·37-s − 12·39-s − 4·41-s − 4·43-s − 2·45-s + 6·47-s + 4·51-s + 12·55-s + 24·57-s − 4·59-s + 12·65-s + 8·67-s + 24·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 1/3·9-s + 1.80·11-s + 1.66·13-s − 1.03·15-s − 0.485·17-s − 2.75·19-s − 2.50·23-s + 3/5·25-s + 1.15·27-s − 1.11·29-s − 2.15·31-s − 2.08·33-s − 0.657·37-s − 1.92·39-s − 0.624·41-s − 0.609·43-s − 0.298·45-s + 0.875·47-s + 0.560·51-s + 1.61·55-s + 3.17·57-s − 0.520·59-s + 1.48·65-s + 0.977·67-s + 2.88·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 132 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 18 T + 257 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.227753749279112102111243207249, −8.227188091734527205532247432425, −7.48651952874481609730131209524, −6.87541005794085960650414758877, −6.46731816068341443294093132825, −6.46084937835739268300197141702, −6.13941561155043078890384790490, −5.80984011499928232161241832332, −5.39924192322974029671107887965, −5.15928931602069801357098402415, −4.36569752157731304172214868487, −3.98357818011455513700172830582, −3.74444333569766820710932935154, −3.57829852184502176415068054402, −2.37369162524291894141534517724, −2.19375348787952797706960541758, −1.61885088638283694657200331975, −1.30254677146395406236854028154, 0, 0, 1.30254677146395406236854028154, 1.61885088638283694657200331975, 2.19375348787952797706960541758, 2.37369162524291894141534517724, 3.57829852184502176415068054402, 3.74444333569766820710932935154, 3.98357818011455513700172830582, 4.36569752157731304172214868487, 5.15928931602069801357098402415, 5.39924192322974029671107887965, 5.80984011499928232161241832332, 6.13941561155043078890384790490, 6.46084937835739268300197141702, 6.46731816068341443294093132825, 6.87541005794085960650414758877, 7.48651952874481609730131209524, 8.227188091734527205532247432425, 8.227753749279112102111243207249

Graph of the $Z$-function along the critical line